Universal theories categorical in power and kappa-generated models

Details Universal theories categorical in power and kappa-generated models Universal theories categorical in power and kappa-generated models

1994-01-15

arxiv.org/abs/math/9401213v1

Ann. Pure Appl. Logic 69 (1994), 27--51

Mathematics

Logic

Scientific paper

We investigate a notion called uniqueness in power kappa that is akin to categoricity in power kappa, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for (uncountable) universal theories T, that if T is kappa-unique for one uncountable kappa, then it is kappa-unique for every uncountable kappa ; in particular, it is categorical in powers greater than the cardinality of T.

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